Abstract
In this chapter we look at cubic equations over a finite field, the field of integers modulo p. We denote this field by \(\mathbb{F}_{p}\). Of course, now we cannot visualize things, but we can look at polynomial equations
with coefficients in \(\mathbb{F}_{p}\) and ask for solutions (x, y) with \(x,y \in \mathbb{F}_{p}\). More generally, we can look for solutions \(x,y \in \mathbb{F}_{q}\), where \(\mathbb{F}_{q}\) is an extension field of \(\mathbb{F}_{p}\) containing q = p e elements. We call such a solution a point on the curve C. If the coordinates x and y of a solution lie in \(\mathbb{F}_{p}\), we call it a rational point.
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Notes
- 1.
We define and study cubic curves that do have complex multiplication in Chapter 6
- 2.
In theory and in practice, there are many methods that are used to check if a number is prime or composite. If you are interested in learning more about this topic, look up “primality testing,” or more specifically the “Miller–Rabin test” and the “Agrawal–Kayal–Saxena (AKS) test.”
- 3.
In practice, one can use more efficient bookkeeping to avoid storing the whole table of values; see Exercise 4.23.
- 4.
If a and n are not relatively prime, then Fermat’s little theorem cannot be used. But in the unlikely event that \(\gcd (a,n) > 1\), the gcd is already a non-trivial factor of n.
- 5.
If the two prime factors of N = pq are of approximately the same size, then the number field sieve is faster than the elliptic curve factorization method described in Section 4.4. But if p is significantly smaller than q, then the elliptic curve method may be faster, since it takes roughly \(e^{c\sqrt{\log p}}\) steps to factor N.
- 6.
We will always assume that P and Q are chosen so that there is such an m.
- 7.
These are all actual real-world applications, although some use something called a digital signature, rather than a public key cryptosystem.
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Silverman, J.H., Tate, J.T. (2015). Cubic Curves over Finite Fields. In: Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-18588-0_4
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